R. Brauer's problem of the group-theoretic characteristic of the number of blocks with a given defect is considered in this paper. The following theorems are proved:
Theorem 1. The number of blocks with defect $d$ of a group $G$ does not exceed the number of $p$-regular non-nilpotent classes with defect $d$.
The theorem is a reinforcement of a result of Brauer — Nesbitt. An example is given showing that this estimate is not always attained.
Corollary 3. Let $G$ be a finite group of order $p^aq ((p, q) = 1$, $p$ is a prime number) containing a normal subgroup $H$ of the order $p^{\gamma}q (0 < \gamma < a)$, some sylow $p$-subgroup of which is a normal subgroup of $H$. Then the number of blocks with defect $d$ coincides with the number of $p$-regular non-nilpotent classes of $G$ with the same defect.
Theorem 3. There exist no blocks with zero defect in the group $G$ if, and only if, all classes with defect zero are nilpotent.
A new proof is also presented for Brauer's theorem on the number of blocks with a maximum defect.

Vibrations of a horizontal elastic shaft with a fixed heavy disk on it is described by a system of three nonlinear differential equations of the second order. A. Stodola suggested a particular solution of this system which corresponds to the second critical speed, but a purely theoretical analysis, of the stability of the particular solution suggested by A. Stodola has not been carried out.
An attempt to find a solution of this problem is made in this article. In his analysis the writer uses the method of calculating the logarithm of the monodromy matrix suggested by [5], [7, a], [4].
In our case it enabled us
a)to state the necessary and sufficient conditions of stability,
b)to calculate the zone of stability in the space of parameters.

Let there be a non-singular matrix $C_r$ for every natural $r$, equation (1) to be satisfied for every vector $T$. All scalar functions $H(T)$ with these properties are described, certain assumptions as to their continuity being made.
This problem is suggested by one probabilistic problem.

The method of approximate solution of equations, based on the assumption of the local nature of the perturbations, is applied to an investigation of the convectional stability in the case of various models accepted for thedescription of plasma: magnetic hydrodynamics, two-fluid hydrodynamics, (the adiabatic and non-adiabatic case), kinetic equations for rarefied plasma (without taking into account collisions). The stability conditions obtained do not depend on the form of the perturbations and are, consequently, of a
universal nature. The results obtained for different plasma models are compared.

The author shows that every rapidly converging iterative process $F(x)$, constructed by a given iterative process $f(x)$, in a certain vicinity of a stationary point, where $F(x)$ and $f(x)$ are assumed to be sufficiently smooth, can be presented in the form (4).
The necessary and sufficient conditions of convergence of an iterative sequence generated by $F(x)$ are found. It is shown that any interval containing a stationary point of the iterative process $f(x)$, in which (8) occurs, is a region of attraction of this stationary point as a stationary point of the process $F(x)$.